You definitely can kill a plant's root system with too much fertilizer, but I'm betting the main problem is not enough light. Sinningias (common name gloxinia, a different species entirely actually) need a lot of light and people often have trouble growing them in the house because they need so much more than other plants in the same (gesneriad) family like African Violets and Streps.

The roots come from the tuber, so if the plant is wobbly then the questions is: Does the tuber wobble too. I suspect the condition is caused by the weak spot where the stem comes up from the tuber. There is nothing that can be done for this now. It probably needs more light, both to grow more compactly and to set lots of buds. But, at least with your feeding you are building up the tuber. Its not too late though to flower and enjoy the plant as it is. I would put in some small stakes, such as bamboo skewers to support the growth, so that it doesn't lean or, worse, break off. If there are buds in the crown then it is going to flower. If not, then move it to a brighter location. Florist Gloxinias are easy to root and propagate. You can root crowns or leaves quite easily in an enclosed environment and in good light.

Jon

Rosemary,
Leginess seems to be a characteristic of florist gloxinias. I have read on the AGGS forum that the best thing to do (besides good light)is pot the soil up to the lowest set of leaves. If the first baby leaves are still there, you can strip them and pot up the the next lowest. The newly potted portion will root given high humdity. I have learned this after removing two plants from the tuber. (Mine went into dormancy and three little guys popped up instead of one.)
-CG

Here is a link that might be useful: AGGS Gloxinia Forum

thanks so much for all the info. very helpful stuff!